We propose a new class of time frequency (TF) symbols covariant to time shi
fts and frequency shifts on a random process. The new TF symbols are useful
for analyzing linear time-varying systems or nonstationary random processe
s, and they are defined as TF-smoothed versions of the narrowband Weyl symb
ol. We derive kernel constraints for the new TF symbols to satisfy the unit
arity property and the quadratic form, We also propose a new class of TF sy
mbols covariant to time shifts and scale changes on a random process. These
new TF symbols can be interpreted as affine-smoothed versions of the narro
wband Weyl symbol or of the wideband P-o-Weyl symbol.